The H-join of arbitrary families of graphs - the universal adjacency spectrum

Abstract

The H-join of a family of graphs G={G1,…,Gp}, also called generalized composition, H[G1,…,Gp], where all graphs are undirected, simple and finite, is the graph obtained by replacing each vertex i of H by Gi and adding to the edges of all graphs in G the edges of the join Gi∨Gj, for every edge ij of H. Some well known graph operations are particular cases of the H-join of a family of graphs G as it is the case of the lexicographic product (also called composition) of two graphs H and G, H[G]. During long time the known expressions for the determination of the entire spectrum of the H-join in terms of the spectra of its components (that is, graphs in G) and an associated matrix, related with the main eigenvalues of the components and the graph H, were limited to families G of regular graphs. In this work, with an approach based on the walk-matrix, we extend such a determination, as well as the determination of the characteristic polynomial, to the universal adjacency matrix of the H-join of families of arbitrary graphs. From the obtained results, the eigenvectors of the universal adjacency matrix of the H-join can also be determined in terms of the eigenvectors of the universal adjacency matrices of the components and an associated matrix.